The LQG Hilbert space \(L_2(\mathrm {SU}(2), \mathrm {d}g)^{|L|}//\mathrm {SU}(2)^{|N|}\)  is standardly understood as spanned by the spin network states labelled by spins. The quantization of the spinorial phase space allows us to span the same Hilbert space by the coherent or scaleless spin network states labelled by spinors. To this end, we also need to introduce a Haar measure. It is given by the Haar measure \(\mathrm {d}\mu (z)\)  of the Bargmann space \({\mathcal {F}}_2=L_2^{\rm hol}({\mathbb {C}}^2,\mathrm {d}\mu )\) , the space of holomorphic squared integrable functions, over spinor [1]}:
\(\mathrm {d}\mu (z):=\frac{1}{\pi ^2}e^{-\left<z|z\right>}\mathrm {d}z^0 \mathrm {d}z^1\,.\) 
